Are you sure about the first rule? It doesn't seem necessary according to https://oeis.org/A014575.
The problem is really fun.
Since I created the problem series and want the Functions I badge, I thought I'd provide the solution that incorporates vampire_factor and are_fangs. Not a great score, but readable!
Sorry, I mean the Magic Numbers badge.
Determine whether a vector is monotonically increasing
Roll the Dice!
Project Euler: Problem 5, Smallest multiple
Project Euler: Problem 6, Natural numbers, squares and sums.
Back to basics 9 - Indexed References
Multivariate polynomials - emulate symbolic form
Negation the hard way
Multivariate polynomials - convert monomial form to array
Product of two multivariate polynomials
Outer product of multiple vectors
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